Model Order Reduction and Control Polynomial Approximation for Well-Control Production Optimization

Abstract

The objective of this paper is to reduce the computational effort in reservoir flooding optimization problems by a combination of different optimization parametrization methods and model order reduction techniques. We compare three different parametrization methods that reduce the cardinality of the original infinite set of control-decision variables to a finite set. The three methods include a traditional piece-wise constant (PWC) approximation, a polynomial approximation by Chebyshev orthogonal polynomials and a piece-wise polynomial approximation by cubic Spline interpolation. The Proper Orthogonal Decomposition with Discrete Empirical Interpolation Method (POD-DEIM) accomplishes the reduced order modeling (ROM).. We compare a gradient-free global stochastic search approach and a gradient-based local search approach. We used Particle Swarm Optimization (PSO) as a gradient-free algorithm and Interior-Point Optimization (IPOPT) with L-BFGS method as a gradient-based algorithm. First, we compare the performances of the three parametrization methods solved by each optimizer, using fine scale simulations for an increasing level of parametrization refinement. Then, in the second part of this paper, we combine the parametrization methods with the reduced modeling workflow. For a given level of parametrization refinement, we compare the performance of each parametrization method coupled with POD-DEIM, and solved by each optimizer. In this part, we introduce an online training procedure, where the first optimization iteration is used to construct the snapshot matrix. The results demonstrate how refining the control approximation with more decision variables per well lead to better NPV values, but with a higher computational cost. The best NPV was achieved using the highest refining level with Chebyshev polynomial approximation. Both polynomial and piece-wise polynomial approximations served as better training sets for POD-DEIM leading to a more accurate and fast reduced model. With the strategy proposed, POD-DEIM showed the best optimization accuracy for Chebyshev polynomial with the gradient-free optimizer, thus permitting the use of the model reduction methodology for global-stochastic search methods. However, the gradient-based approach seems to consistently outperform the gradient-free approach in terms of NPV and number of iterations for the cases shown.

Non-Intrusive Parametric Model Order Reduction With Error Correction Modeling for Changing Well Locations Using a Machine Learning Framework

“…Most of the PMOR methods for well control optimization that have been relies on the simulator source code. Methods like POD-TPWL [11] are intrusive to some extent as it requires access to Jacobian and residual matrices, that is not easily available for commercial simulators and POD-DEIM [17] is highly intrusive to the source code. One of the non-intrusive methods that has been used for changing well controls is DMD [37] and DMDc [22].…”

“…This method proved to be more accurate as compared to TPWL but the overhead time increases significantly because it involves third-order matrix-tensor products. Another such method called Discrete Empirical Interpolation Method (DEIM) [14] was also introduced applied in combination with POD for changing well control optimization problem [15,16,17], which can provide higher accuracies as it aims at reducing the dimensions of discretized parametric PDEs by computing the non-linear terms at discrete locations in the spatial domain and then interpolating them to the rest of the locations using projection based interpolation. However, less speed-ups have been reported for POD-DEIM as compared to POD-TPWL as a result of linearization performed in the latter and it is also more invasive with respect to the simulator as compared to POD-TPWL.…”

A State-of-the-Art Literature Review on Capacitance Resistance Models for Reservoir Characterization and Performance Forecasting

“…In this parameterization, it is important to have an efficient way to span the solution space considering trajectories that are logistically viable. The results obtained by Sorek et al [96] suggest that the use of Chebyshev orthogonal polynomials as the basis for such parameterization results in faster convergence and higher NPV with smooth trajectories for well control, which is desirable from an operational perspective.…”

A globally convergent method to accelerate large-scale optimization using on-the-fly model hyperreduction: application to shape optimization

“…Most of these methods either address linear problems [47,32,45] or do not incorporate hyperreduction [50,49]. The work that does directly address nonlinear problems and incorporate model hyperreduction [39,37,30] are not currently equipped with global convergence theory.…”